As you mention it, it sounds like your manager is stopping you from making fair usage of the company policy. It's bad.
Reach out to the HR, immediately. This is a malpractice that needs to stop. Simply put, your manager is asking you to work for free to show up the profit in the balance sheet. That's unfair and not correct.
As you mentioned, your manager already mentioned about the "profit" and did not inform you to use the overtime policy - it's understood that he is not doing this by mistake, this is a deliberate attempt.
Detail the situation in writing to HR. Mention the nature of the work request and ask what alternative could be arranged to see that you are properly compensated.
Either ask him a compensatory off on weekday for working on Saturday (There's no provision for compensatory off in our office rules, but he can give a comp off unoficially)
I'd not take that path. If something is not in the rulebook, don't use it. It may or may not be honored, if the current arrangement changes.
- Also, if it's you who asks for it, you can be in a tricky situation in case there's a problem later on.
- In case it's your manager proposing this out-of-the-book arrangement, it's less problematic for you, but given that the process is still outside company policy, in case of any conflicts, you'll lose all the comp-offs or similar.
Convince him to let me charge the Saturday in timesheet.
You already tried that and received a refusal - it's very less likely that the answer will change on a second request. Better use the proper channel.
He says we cannot show Saturday work to our US offices (US office manages all the finance and administration) and cannot charge it to the time sheet since it will affect the project profits
That clearly indicates, your manager is incapable of managing. It's your managers (one of the many) responsibility to take care of the interests of the employees' working with them, not try to take advantage. In case you see this to be a company-wide policy (or,location-based malpractice), rather than a one-off case with your project / manager, it's time to find a new job.
In the end, if your time is not honored, it's not worth the time and effort.